A novel extension of half-logistic distribution with statistical inference, estimation and applications

In the present study, we develop and investigate the odd Frechet Half-Logistic (OFHL) distribution that was developed by incorporating the half-logistic and odd Frechet-G family. The OFHL model has very adaptable probability functions: decreasing, increasing, bathtub and inverted U shapes are shown for the hazard rate functions, illustrating the model’s capacity for flexibility. A comprehensive account of the mathematical and statistical properties of the proposed model is presented. In estimation viewpoint, six distinct estimation methodologies are used to estimate the unknown parameters of the OFHL model. Furthermore, an extensive Monte Carlo simulation analysis is used to evaluate the effectiveness of these estimators. Finally, two applications to real data are used to demonstrate the versatility of the suggested method, and the comparison is made with the half-logistic and some of its well-known extensions. The actual implementation shows that the suggested model performs better than competing models.


Reliability function
The mathematical expressions for reliability function of OFHL(α, θ ) distribution is calculated as

Hazard rate function
The associated hazard rate function of OFHL(α, θ ) distribution takes the form Figure 1 visually illustrates the PDF of OFHL model across various values of parameters α and θ .This depic- tion showcases the versatility of the OFHL model, exhibiting PDFs that can be unimodal, right-skewed, symmetric, or even demonstrate an increasing density function.These variations portray the model's adaptability in capturing diverse data patterns related to lifetime distributions.
Figure 2 depicts the graphical representation HRF for distinct parameter values of the OFHL model.This graphical representation demonstrates the flexibility of the OFHL model in modeling different types of lifetime data by displaying HRFs that can be decreasing, increasing, exhibit a bathtub-shaped pattern, or even an inverted bathtub-shaped pattern.This flexibility highlights the model's capability to capture a wide range of behaviours observed in lifetime data analysis (Fig. 3).

Series expansion of the OFHL model
A useful series expansion of the PDF and CDF of the OFHL model is provided in this section.
The series expansion of e −y is derived by using Taylor series expansion as given by www.nature.com/scientificreports/ The binomial series expansion of (1 − x) −n is derived by using the binomial theorem and Taylor series expan- sion.The expansion is derived as Then, it follows that, using Eq. ( 9) and Eq.(10), we can expand (−1) ξ y ξ ξ ! .

Basic properties
This part is devoted to derive some of the basic properties of the OFHL(α, θ ) model such as quantile function, ordinary moments, incomplete and conditional moments and order statistics.

Quantile function
By inverting the CDF of the OFHL(α, θ ) model, the expression for quantile function Q(u) of the random variable X is calculated as where u is the uniform random variable defined on a unit interval [0,1].As a result, the median (M) of the OFHL(α, θ ) model is computed by setting u = 0.5 , we get

Moments and moment generating function
If X follows OFHL(α, θ) model, then the rth moment about origin (raw moments) can be evaluated by extending the PDF given by Eq. ( 13) Using Integration via substitution method in Eq. ( 14), we perform the following operations: Let, Thus, On simplification, we get www.nature.com/scientificreports/ The moment generating function of OFHL(α, θ ) model utilizing the Maclaurin series is mentioned as Thus, by using Eq. ( 15), the expression for moment generating function is computed as

Incomplete and conditional moments
If X belongs to OFHL(α, θ) distribution, then the rth incomplete moment is given by Using the PDF (13), we can write On simplification, we obtain where, γ (a, b) = a 0 x b−1 e −x dx is the lower incomplete gamma function.
Furthermore, the rth conditional moment, say � r y = E Y r |Y > y is given by Hence, by using (13), r y can be written as On simplication, the final expression comes out to be

Mean residual life and mean waiting time
Suppose X is a continuous random variable having survival function R(x) , the mean residual life function, say π(t) is defined as the expected life of an item after it has reached a certain age t , is given where, and η k,ℓ 2 α(k+1) y 0 x r e −θ x α(k+1)+ℓ dx Vol.:(0123456789) www.nature.com/scientificreports/By using ( 7), ( 17) and (18), π(t) can be written as where, The mean waiting time is very important to analyse the actual time of failure of an already failed item.It represents the amount of time that has passed since an object failed, assuming that the failure occurred within the interval [0, t].The mean waiting time say π(t) , is defined by By using ( 6) and ( 18), the final expression of π(t) comes out to be

Order statistics
In real-world applications incorporating data from life testing studies, order statistics is very important.Assume that X 1 , X 2 , . . ., X n be a random sample with relevant order statistics given by X (1) , X (2) , . . ., X (n) .The CDF of the nth or maximum order statistics, say F n:n (x) , is given as Consequently, the PDF of the nth order statistics, say f n:n (x) , is computed as The CDF and PDF of minimum order statistics is given by and

Parameter estimation of OFHL model
This section is devoted to discuss various estimation methodologies for estimating the unknown parameters of the OFHL model, such as the Maximum Likelihood Estimation (MLE), Anderson-Darling Estimation (ADE), Cramer-von Mises Estimation (CVME), Maximum Product of Spacing Estimation (MPSE), Ordinary Least Square Estimation (OLSE) and Weighted Least Square Estimation (WLSE).

Maximum likelihood estimation
Let x 1 , x 2 , , . . ., x n be a random sample of size n following OFHL model with parameters α and θ , then the loga- rithmic likelihood function is (18) η k,ℓ AB.
Vol:.( 1234567890) www.nature.com/scientificreports/By differentiating (20) with respect to unknown parameters α and θ , the resulting partial derivatives are given by and By setting the above partial derivatives equal to zero, we could calculate the ML estimators α ML and θ ML of the unknown parameters α and θ .Since, the given equations are not in a closed form and cannot be derived analytically.However, R software can be used to get ML estimators of the parameters.

Anderson darling estimation
The Anderson-Darling test Anderson and Darling 23 can be used in place of other statistical tests to identify deviations from normality in sample distributions.The AD estimators denoted by α AD and θ AD of the parameters can be evaluated by minimizing the following function with respect to α and θ , respectively

Cramer-von-Mises estimation
Our choice to utilise minimal distance estimators of the Cramer-von-Mises type was supported by empirical data from Macdonald 24 proving that the bias of the estimator is smaller than that of the competing minimum distance estimators.The Cramer-von-Mises estimators α CVM and θ CVM of α and θ are derived by minimizing the value of the following function

Maximum product of spacing estimation
This approach was first developed by Cheng and Amin 25 as an alternative to ML estimation.The uniform spacing for a random sample of size n taken from the OFHL model can be determined by where D k denotes the uniform spacing, F x (0) = 0 and F x (n+1) = 1.
The MPS estimators α MPS and θ MPS of the unknown parameters α and θ can be obtained by maximizing the following function ( 20) For estimating the unknown parameters, the least square (LS) and weighted least square (WLS) approaches are widely known Swain et al. 26 .Here, the two approaches for parameter estimation of OFHL model are examined.By minimising the following function with respect to α and θ respectively, it is possible to obtain the LS and WLS estimators α and θ of the OFHL distribution By setting n k = 1 , the LS estimators α LS and θ LS can be obtained, while as by setting n k = (n+1) 2 (n+2) k(n−k+1) , we can obtain the WLS estimators denoted by α WLS and θ WLS .

Simulation study
It is not theoretically possible to compare the effectiveness of the introduced estimation methods derived in the previous section for estimating the parameters of the OFHL model.Thus, we undertake a Monte Carlo simulation analysis to identify the top estimation method among the six classical estimation methods.In order to do this, we created 1,000 samples at random of sizes 20, 40, 100, 200 and 400 from the OFHL model for three different sets of parameter values, as shown below: In this simulation study, we evaluate the average values of estimates (AVEs), biases, mean square errors (MSEs) and mean relative errors (MREs).The following mathematical formulas are used to accomplish these objectives: where ϑ = (α, θ) .All the results related to simulation were obtained by using R-Studio software.The results of the simulation are displayed in Tables 1, 2, 3.

Interpretations at the end of the simulation Results
• The absolute biases of α and θ decreases as n rises under all estimation techniques.• As n increases, the MSE reduces for all the methods of estimation, satisfying the consistency criteria.
• In all estimation procedures, as n increases, the discrepancy between estimates and specified parameters decreases.• In terms of MSE, the method of maximum product of spacing estimation outperforms the other methods in the majority of situations.• In light of our analysis and from Table 4, we determine that MPSE performs best (overall score of 23.5) as n approaches infinity in terms of bias, MSE and MRE for the parameter combinations taken into account in our study.• In most instances, the second best performing estimator is MLE (overall score of 27.5) followed by ADE (overall score of 47.5).• The overall position of the remaining estimators is displayed in Table 4.
The overall inference drawn from the simulation results is that as sample size increases, bias, MSE and MRE for all parameters goes on decreasing and eventually will reach to zero.This shows the precision of both the numerical computations for the OFHL parameters and the estimation techniques.

Real data applications
To highlight the significance of the OFHL distribution discussed in Section "The odd Frechet Half-Logistic (OFHL) model", we demonstrate two real applications to assess the adaptability of the subjected model.
Dataset I: The first dataset covers 108 days from 4 March to 20 July 2020 and corresponds to the COVID-19 mortality rate for Mexico.It was previously examined by Almongy et al. 27 Set I : α = 0.25, θ = 0.75.
Furthermore, we fitted the OFHL distribution by utilizing the six estimation procedures and the results are reported in Tables 7 and 8.The estimated PDF, SF, P-P and Q-Q plots of OFHL model for two datasets are contrasted in Figs. 4, 5, 6 and 7 respectively.To sum up, the OFHL model demonstrates that it is the most appropriate model for the two datasets by illustrating how it may be applied in a real-world scenario.
Using the TTT (Total time on test) plot recommended by Aarset 30 , the form of the hazard function of the datasets was assessed and the results demonstrate that both datasets display different shapes of hazard rate.Iftikhar et al. 31 also employed this technique for assessing the graphical overview of hazard rate of the data.It was introduced for the two real data sets in Fig. 3.

Final comments on the data analysis results
• From the Tables 5 and 6, we can infer that our suggested distribution performs better as compared to other competing models.• In dataset I and dataset II, OFHL model has the highest p-value as well as smallest AD, CVM and KS distance.

Conclusion
In the current study, we propose a novel two parameter model, namely two-parameter odd Fréchet Half-Logistic (OFHL) distribution; its mathematical features have been thoroughly described.The OFHL distribution is more adaptable for analyzing lifespan data as compared to other models.The suggested model contains a broad variety of forms, which boosts its flexibility in modeling different types of data, as inferred from the PDF and hazard rate plots.Several conventional estimation approaches, including MLE and five other methods, were used to estimate the unknown parameters of the proposed model.A simulation study with 1000 iterations was conducted to analyse and evaluate the performance of the estimation approaches and it was found that as n increases, the estimated biases, MSEs and MREs of the parameters α and θ under the MPSE estimation approach quickly decreases, demonstrating the effectiveness of the MPSE procedure.Further, the superiority and effectiveness of the suggested model over some of its competitors was further established using real-world data analysis, which demonstrates that the underlying model fits the data more accurately than the other distributions.We anticipate that the findings from this study will be valuable for practitioners in a variety of fields.

Figure 3 .
Figure 3. TTT plots of the Covid-19 and Survival time datasets.

Figure 4 .
Figure 4. (i) Fitted density plot of OFHL distribution.(ii) The fitted survival plot and empirical survival plot of OFHL for Covid-19 dataset.

Figure 5 .
Figure 5. (i) Fitted density plot of OFHL distribution.(ii) The fitted survival plot and empirical survival plot of OFHL for Survival time dataset.

Figure 6 .
Figure 6.P-P plots of the OFHL for Covid-19 and Survival time datasets.

Table 4 .
Partial and overall rankings of all estimation methodologies.

Table 6 .
The MLEs and GoF metrics statistics for Survival time dataset.

Table 7 .
Estimate of Parameters and GoF metrics for Covid-19 dataset using various estimation approaches.

Table 8 .
Estimate of Parameters and GoF metrics for Survival time dataset using various estimation approaches.